Determination of the strong coupling at NNLO from jet production in DIS
aa r X i v : . [ h e p - ph ] M a y Determination of the strong coupling at NNLO from jetproduction in DIS
Daniel Britzger ∗ on behalf of the collaboration † DESY, Notkestr. 85, 22607 Hamburg, GermanyE-mail: [email protected]
A first determination of the strong coupling α s ( m Z ) in next-to-next-to-leading order (NNLO)from inclusive jet and dijet production cross sections in deep-inelastic scattering at HERA ispresented. Data collected by the H1 experiment in the years 1995 to 2007 covering the rangeof momentum transfer 5 . < Q <
15 000 GeV and jet transverse momenta P jetT > . α s ( m Z ) = . exp ( + − ) theo . Further studies on the phenomenological application of the new NNLOcalculations and on fits to the individual data sets are presented. The running of the strong cou-pling is probed in a single experiment over one order of magnitude in the remormalisation scaleand consistency with the QCD expectations is found. XXV International Workshop on Deep-Inelastic Scattering and Related Topics3-7 April 2017University of Birmingham, UK ∗ Speaker. † Work performed by the H1 Collaboration together with V. Bertone, J. Currie, C. Gwenlan, T. Gehrmann, A. Huss,J. Niehues and M. Sutton c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ trong coupling at NNLO from H1 jet cross sections
Daniel Britzger
1. Introduction
The strong coupling constant is one of the least known parameters of the Standard Model (SM)and a precise knowledge is of crucial importance for precision physics and searches for physicsbeyond the SM at the LHC. Cross sections for jet production in deep-inelastic electron-protonscattering (DIS) are directly sensitive to the strong coupling constant α s ( m Z ) already in leadingorder in perturbative QCD (pQCD) as these measurements are performed in the Breit frame ofreference. The cross section calculations are performed in next-to-next-to-leading order (NNLO)accuracy, where the cross section predictions are obtained with the program NNLO JET [1, 2].
2. Methodology
Cross sections for jet production in ep collisions have been measured by the H1 experiment atHERA at di ff erent center-of-mass energies and for di ff erent kinematic regions. Here, jet and dijetcross sections taken during the years 1995 to 2007 [3, 4, 5, 7, 6] are considered. Consistent to alldata sets, jets are defined using the k t jet-algorithm with a parameter of R =
1, and jets are requiredto be contained in the pseudorapidity range − < η jetlab < . ff erentially as a function ofthe photon virtuality Q and jet transverse momentum P jetT , and dijet cross sections as a functionof Q and the average transverse momentum of the two hardest jets, h P T i . A brief summary ofthe employed measurements and the kinematic range of the observables is given in table 1. The Data set √ s int. L DIS kinematic Inclusive jets Dijets[Ref.] [GeV] [pb − ] range n jets ≥ < Q < < P jetT < . < h P T i < < Q < < P jetT < < h P T i < < Q < < P jetT < − HERA-II [6] 319 290 5 . < Q < . < P jetT < < h P T i < < Q < < P jetT < < h P T i < Table 1:
Summary of the kinematic ranges of the inclusive jet and dijet data taken by the H1 experiment. data sets are separated into di ff erent data taking periods and two Q -regions, where the scatteredlepton is identified in di ff erent experimental devices. In case of the dijet cross sections, regions ofthe phase space exhibiting an infrared sensitivity due to ‘back-to-back’ topologies are avoided byimposing asymmetric cuts on the transverse momenta of the two leading jets.The predictions are calculated as a convolution of parton density functions (PDFs) and a par-tonic cross section. Both these components exhibit a dependence on α s ( m Z ) and their impact onthe results are assessed below. The partonic cross section has its α s -dependence explicit as it iscalculated in terms of a perturbative expansion in orders of α ( n ) s . The α s -dependence of the PDFs isgiven by the factorisation theorem and where it originates from the QCD splitting kernels and the β -functions. Once a PDF is determined for a given value of α s ( m Z ) it can be translated to any othervalue of α s ( m Z ) by an integration step. This translation defines the α s -dependence of the PDF.An equivalent solution to this explicit integration is obtained by evaluating the PDFs at a suitablevalue of µ F , which depends on α s ( m Z ), thus taking full benefit of the factorisation theorem [8].1 trong coupling at NNLO from H1 jet cross sections Daniel Britzger
The PDF parameterisation is based on the NNPDF3.0 PDF set [9], which was determined for avalue of α s ( m Z ) = . ff ects. The renormalisation and factorisation scales are chosen to be µ R = µ F = Q + P , where P T denotes P jetT in case of inclusive jet and h P T i for dijet cross sections.The value of the strong coupling constant is determined in a fit of these NNLO calculations tothe H1 jet data, where the α s -dependencies in the predictions, both in the partonic cross sections andin the PDF, are taken into account. The NNLO coe ffi cients are stored in the fastNLO format [10] inorder to allow for a repeated calculation with di ff erent values of α s ( m Z ) and di ff erent PDF sets. Thefit χ -definition accounts for experimental, hadronisation and PDF uncertainties. Correlations ofsystematic uncertainties and statistical correlations of the data are considered. The uncertainties onthe resulting value of α s ( m Z ) due to experimental and theoretical sources are estimated. The PDFand hadronisation uncertainties are obtained by repeating the fit with these uncertainties excludedin the fit and comparing the resulting fit uncertainty. The scale uncertainty is estimated by repeatingthe fit with scale factors of 0.5 and 2. The ‘PDFSet’ uncertainty is obtained as half of the maximumdi ff erence of the results from fits using alternatively the ABMP, CT14, HERAPDF2.0, MMHT orNNPDF3.0 PDF set, and the ‘PDF α s ’ uncertainty is estimated as half of the di ff erence of the resultsobtained from fits using PDFs which were determined with α s ( m Z )-values di ff ering by 0.004. ) f r o m f i t Z ( m s f α GeV < 15 µ H1 inclusive jets % C.L. GeV < 15 µ H1 dijets χ Fits using in exp. uncertainty onlyexp., had. & PDF unc. ) from fit Z (m σ s α ) f r o m f i t Z ( m s f α GeV > 15 µ et al. (preliminary) H1 ) from fit Z (m σ s α GeV > 15 µ Figure 1:
Contours at a confidence level of 68 %for fits where the two appearances of α s ( m Z ) in thecross section calculation are identified separately.The upper and lower pads show results from datapoints with µ R smaller or greater 15 GeV. Thedashed contours indicate fits where the PDF uncer-tainty is not considered in the χ -calculation. ) f r o m f i t t o H j e t s Z ( m s α -fit NNLO s α Scale choice -fit NNLO s α Scale uncertaintyNLO µ Scale choice ( Q T P T P + Q T P + Q T P + Q T P + Q do f n / χ et al. (preliminary) H1 = dof n ) µ Scale choice ( Q 〉 T P 〈 〉 T P 〈 + Q 〉 T P 〈 + Q 〉 T P 〈 + Q 〉 T P 〈 + Q = dof n Figure 2:
Values of α s ( m Z ) obtained from fits toinclusive jet or dijet cross sections obtained for dif-ferent definitions of the renormalisation and factori-sation scales. The lower pads show the values of χ / n dof of the fit. The open circles display resultsobtained using NLO matrix elements. The verticalerror bars indicate the scale uncertainty.
3. Results
The sensitivity of the data to α s ( m Z ) is studied in fits with two free parameters representingthe two α s contributions to the calculation, assuming those can be chosen independently, i.e. one2 trong coupling at NNLO from H1 jet cross sections Daniel Britzger parameter for the PDFs, α fs ( m Z ), and another parameter for the hard coe ffi cients, α ˆ σ s ( m Z ). The fitsare performed using inclusive jet or dijet cross section measurements, with data points below orabove the renormalisation value of 15 GeV. The contours displaying the 68 % confidence levelof the fitted results are displayed in figure 1. The two α s ( m Z ) values determined in the fit areconsistent, while the sensitivity to α s ( m Z ) of the hard coe ffi cients outperforms the one of the PDF.The two fit parameters are negatively correlated, resulting in an increased sensitivity for fits usinga common α s ( m Z ).Fits are also performed employing alternative definitions for the renormalisation and factorisa-tion scales. The resulting α s -values and related values of χ / n dof for the individual fits are displayedin figure 2 for fits to inclusive jet and to dijet cross sections. The results obtained with alternativescale choices are typically covered by the scale uncertainty. Choosing µ R = µ F = Q is disfavored,presumably because this scale is not su ffi ciently related to the dynamics of jet production. For com-parison the fits are also repeated with hard coe ffi cients calculated in NLO accuracy only. Thesecalculations typically yield higher values of χ / n dof of the fits and the scale choice has a higherimpact on the NLO results. These observations emphasize the improved perturbative convergenceof the NNLO calculations as compared to NLO accuracy. ) Z (m s α [2016] World average [NNLO]
H1 jets [NNLO]
H1 dijets [NNLO]
H1 inclusive jets HERA-II high-Q HERA-II low-Q HERA-I low-Q GeV high-Q [all NNLO]
H1 dijets HERA-II high-Q HERA-II low-Q HERA-I high-Q HERA-I low-Q GeV high-Q [all NNLO]
H1 inclusive jets et al. (preliminary) H1 Figure 3:
Summary of the values of α s ( m Z ) ob-tained from fits to the individual data sets and fromfits to multiple data sets. The inner errors bars indi-cate the experimental uncertainty and the outer errorbars the total uncertainty. ) R µ ( s α World average 2016 [NNLO]
H1 jets [NNLO+NLLA+K]
JADE 3-jet rate ) [NNLO] et al. (Dissertori, ALEPH y [NNLO] OPAL y [NNLO] tCMS t
LO] [N GFitter EW fit et al. (preliminary) H1 [GeV] R µ ) Z ( m s α Figure 4:
Values of α s ( m Z ) obtained from fits to‘H1 jets’ data points with similar values of µ R (fullcircles) in comparison to values from other experi-ments and processes, where all values are obtainedat least in NNLO accuracy. The fitted values of α s ( m Z ) are translated to α s ( µ R ) using the solutionof the QCD renormalisation group equation as theyalso enter the calculations. The inner error bars dis-play the experimental uncertainties and the outer er-ror bars indicate the total uncertainties. The values for α s ( m Z ) obtained from fits to the individual data sets are displayed in figure 3 andcompared to the world average value of α s ( m Z ) = . ± . trong coupling at NNLO from H1 jet cross sections Daniel Britzger
A fit to all H1 jet cross section data (denoted ‘H1 jets’), where however the HERA-I dijet crosssections are excluded from the fit because their statistical correlations to the inclusive jets are notprecisely known, yields a value of χ / n dof = .
03 for 203 data points and the value of the strongcoupling constant α s ( m Z ) is determined to α s ( m Z ) = . exp (3) had (6) PDF (12)
PDF α s (2) PDFset ( + − ) scale . This is consistent with the world average and with fits of the individual data sets.The running of the strong coupling constant as a function of the renormalisation scale µ R , isstudied by repeating the fit for groups of data points with comparable values of µ R . The resultingvalues of α s ( m Z ) and α s ( µ R ) are displayed at a representative value µ R for the given range in fig-ure 4. The results confirm the expectations from the QCD renormalisation group equation withinthe accessible range in µ R of approximately 7 to 90 GeV. The α s -values are also compared to α s -determinations at NNLO in other reactions at similar scales and consistency is found.
4. Summary and conclusion
The strong coupling constant is determined in a fit of new next-to-next-to-leading order (NNLO)QCD predictions to inclusive jet and dijet cross section measurements by the H1 experiment as α s ( m Z ) = . exp ( + − ) theo [8], which is in consistency with the world average value. The run-ning of the strong coupling constant is probed over one order of magnitude and consistency with theQCD expectation is found. The NNLO calculations reduce significantly the dominating theoreticaluncertainty in comparison to previously available NLO calculations. The experimental uncertaintyis reduced by considering the entire set of inclusive jet and dijet cross section measurements by theH1 experiment. References [1] J. Currie, T. Gehrmann, and J. Niehues,
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